Diagrams, week 6

This week, we were discussing definitions. In particular, we were trying to decide whether we needed to include the measures of the congruent angles in our definition of a rectangle, or whether simply stating that they are congruent would be good enough.

That is, can we define a rectangle as an equiangular quadrilateral? Or do we need to specify a quadrilateral with four 90° angles?

This led to discussion of whether it is possible for a quadrilateral to have all four of its angles obtuse.

One student argued “No”. She argued that a rhombus typically has two obtuse angles and two acute angles; if you try to make the acute ones obtuse, the obtuse ones become acute.

The class accepted that argument (much to my chagrin).

So I stepped in, saying that I wasn’t convinced. Specifically, I said:

You are arguing that a quadrilateral can’t have four obtuse angles by showing that a rhombus can have at most two right angles. But I don’t think that’s true of quadrilaterals in general. I think it may be possible for a quadrilateral to have three obtuse angles.

If that’s possible, then your argument doesn’t show what we all seem to think it shows. And maybe four obtuse angles is possible.

A few minutes later, a student produced the diagram above on the board. A quadrilateral with three obtuse angles. Furthermore, we all agreed that we could imagine tweaking things so that all three of these angles would be congruent.

What have we learned? That we still don’t know whether we need to state that the congruent angles in a rectangle are right.

3 responses to “Diagrams, week 6”

Can you stress to your students that we like to have definitions which make sense not only in specific cases, but which can be used universally? For example, we are happy with the term “regular pentagon” to describe a specific 5-sided polygon, and that description would be universally understood Stating instead that we have a “pentagon where all angles are 108 degrees” is not only awkward, it comes off as a bit silly and superfluous. The fact that all angles in a regular quadrilateral are 90 degrees is not an essentially part of the definition, but can be instead characterized as a corollary of the definition.

Bob – I agree with you about the pentagon, but with rectangles we are fighting against two different things: students familiarity with rectangles and the importance and prominence of right angles. Of course, defining a rectangle as an equiangular quadrilateral has a certain elegance to it, but I think it will take a long time to lead students there. This is especially true since geometry is one of the first times students bump into the difference between the definition of a mathematical object and its properties.

You both raise interesting points, and they are things that have come up in class for sure. I think Bob‘s issue is one of elegance. Elegance is desirable, but not really essential to mathematical understanding. I see it as a much finer point than much of what we’re working on. The essential piece for us was whether the two proposed definitions are equivalent. That’s where the interesting mathematics can be learned. The rest is aesthetics, which (as always) is a matter of taste.

I’m working on writing up the whole curricular sequence that has led the class to this point. More details around each of your points will surely emerge as I do that.